Let G be a finite group and R a strongly G-graded ring. The question of when R is semisimple (meaning in this paper semisimple artinian) has been studied by several authors. The most classical result is Maschke’s Theorem for group rings. For crossed products over fields there is a satisfactory answer given by Aljadeff and Robinson [3]. Another partial answer for skew group rings was given by Al...

The theory of minimal types for representations of complex semisimple Lie groups [K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, Ann. of Math. (2) 85 (1967), 383-429, Chapters 1, 2 and 3] is reformulated so that it can be generalized, at least partially, to real semisimple Lie groups. A rather complete extension of the complex theory is obtained for the semisimple Lie groups of real r...

We define global dimension and weak dimension for the structured ring spectra that arise in algebraic topology. We provide a partial classification of ring spectra of global dimension zero, the semisimple ring spectra of the title. These ring spectra are closely related to classical rings whose projective modules admit the structure of a triangulated category. Applications to two analogues of t...

(1.5) Proposition Let R be a semisimple ring. Then R is isomorphic to a finite direct product ∏s i=1 Ri, where each Ri is a simple ring. (1.6) Proposition Let R be a simple ring. Then there exists a division ring D and a positive integer n such that R ∼= Mn(D). (1.7) Definition Let R be a ring with 1. Define the radical of R to be the intersection of all maximal left ideals of R. The above defi...

This paper studies the representations of semisimple Lie algebras, with care given to the case of sln(C). We develop and utilize various tools, including the adjoint representation, the Killing form, root space decomposition, and the Weyl group to classify the irreducible representations of semisimple Lie algebras.

We present a simplified version of Tits’ proof of the classification of semisimple algebraic groups, which remains valid over semilocal rings. We also provide explicit conditions on anisotropic groups to appear as anisotropic kernels of semisimple groups of a given index.

We prove that if L = lim ←−Ln (n ∈ N), where each Ln is a finite dimensional semisimple Lie algebra, and A is a finite codimensional ideal of L, then L/A is also semisimple. We show also that every finite dimensional homomorphic image of the cartesian product of solvable (nilpotent) finite dimensional Lie algebras is solvable (nilpotent). Mathematics Subject Classification: 14L, 16W, 17B45

Riemannian and pseudo-Riemannian symmetric spaces with semisimple transvection group are known and classified for a long time. Contrary to that the description of pseudo-Riemannian symmetric spaces with non-semisimple transvection group is an open problem. In the last years some progress on this problem was achieved. In this article we want to explain these results and some of their applications.

Introduction Lecture 1. Algebraic properties of correlators in 2D topological field theory. Moduli of a 2D TFT and WDVV equations of associativity. Lecture 2. Equations of associativity and Frobenius manifolds. Deformed flat connection and its monodromy at the origin. Lecture 3. Semisimplicity and canonical coordinates. Lecture 4. Classification of semisimple Frobenius manifolds. Lecture 5. Mon...

for two algebras $a$ and $b$, a linear map $t:a longrightarrow b$ is called separating, if $xcdot y=0$ implies $txcdot ty=0$ for all $x,yin a$. the general form and the automatic continuity of separating maps between various banach algebras have been studied extensively. in this paper, we first extend the notion of separating map for module case and then we give a description of a linear separa...